EXCHANGE 


ELECTROMAGNETIC 

OSCILLATIONS   FROM  A 

BENT  ANTENNA 


ROBERT  CAMERON  COLWELL 


/ 


ELECTROMAGNETIC   OSCILLATIONS 
FEOM  A  BENT  ANTENNA 


A  DISSERTATION 

PRESENTED  TO  THE 

FACULTY  OF  PRINCETON  UNIVERSITY 

IN  CANDIDACY  FOR  THE  DEGREE 

OF  DOCTOR  OF  PHILOSOPHY 


BY 
ROBERT  CAMERON  COLWELL 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER. HA. 

1920 


Accepted  by  the  Department  of 
Physics,  February,  1918 


'.   '.;',    !f,'r  't     ;    -', 


ELECTROMAGNETIC  OSCILLATIONS  FROM  A  BENT 

ANTENNA 

The  purpose  of  this  investigation  is  to  find  the  mathematical 
equations  for  the  electromagnetic  oscillations  from  a  bent  an- 
tenna, which  is  known  to  send  out  directed  waves.  The  method 
used  is  that  of  Pocklington1  which  has  been  developed  by  M. 
Abraham2  and  particularly  by  G.  W.  Peirce3  who  has  recently 
published  a  remarkable  research  on  the  radiation  resistance  of  a 
flat  top  antenna.  This  article  is  based  upon  the  work  of  Peirce 
and  Abraham,  but  the  equations  are  worked  out  for  the  funda- 
mental and  not  for  the  forced  vibration.  The  application  of  the 
formula  is  new. 

The  following  assumptions  are  made: 

1.  That  any  antenna  may  be  considered  to  be  made  up  of  a 
large  number  of  Hertzian  doublets  placed  end  to  end. 

2.  That  the  earth  is  a  perfectly  conducting  plane. 

3.  That  the  waves  propagated  high  into  the  air  eventually 
return  to  the  earth.     The  reason  for  this  assumption  will  be 
shown  in  the  section  dealing  with  the  horizontal  part  of  the 
antenna. 

Let  a  flat  top  antenna  have  a  vertical  part  h  and  a  horizontal 
part  d,  Fig.  1.  It  will  be  necessary  to  discuss  the  effect  of  the 


EARTH'S  SURFACE 


FIG.  1 

radiation  from  this  antenna  in  three  parts. 
I.  The  radiation  due  to  the  vertical  part  h  and  its  image  —  h. 

1  Pocklington  H.  C.,  Camb.  Phil.  Soc.  Proc.,  1898,  p.  325. 

2  Abraham,  Theorie  der  Electrizitat,  Vol.  II. 

3  G.  W.  Peirce,  "Radiation  Characteristics  of  an  Antenna,"  Proc.  Am. 
Acad.  Arts  and  Sciences,  Vol.  52,  No.  4,  October,  1916. 

3 


ELECTROMAGNETIC   OSCILLATIONS 


II.  The 'radiation  from  the  horizontal  part  d  and  its  image  —  d. 
III.  The  mutual  action  between  the  vertical  and  horizontal  parts. 

I.   RADIATION  FROM  THE  VERTICAL  PART 
Let  the  point  where  the  vertical  part  enters  the  ground  be  the 
origin  of  co-ordinates,  and  choose  the  axes  as  shown  in  Fig.  2 


FIG.  2 

where  the  x  axis  is  parallel  to  the  direction  in  which  the  free 
end  of  the  horizontal  antenna  points  and  the  vertical  antenna 
coincides  with  the  direction  OZ.  The  azimuth  and  co-latitude 
angles  have  their  usual  designations.  Let  P  be  any  point  in 
space  whose  polar  co-ordinates  are  r,  <f>,  6,  where  r  is  very  great 
compared  to  the  height  h  of  the  vertical  part  of  the  antenna. 
P  lies  at  a  distance  r  from  the  point  Pr  which  is  the  position  of 
one  of  the  Hertzian  doublets  postulated  in  the  first  assumption. 
The  effect  of  the  doublet  at  Pf  on  the  point  P  is  given  by  the 
theory  of  Hertz1 

CD 


where  f(t)  =  the  moment  of  the  doublet  edz,  r  —  length  PPf  and 
V  =  velocity  of  light.  E  is  expressed  in  electrostatic  and  H  in 
electromagnetic  units;  rQ  and  6  appear  in  place  of  r  and  6',  as  is 
legitimate  because  of  the  great  magnitude  of  r  in  comparison 
with  OP. 

If  the  total  length  of  the  antenna  is  /  =  h  +  d  there  must  be 
a  node  of  current  at  +  /  and  —  /.     For  the  fundamental  vibra- 

1  Hertz,  Electric  waves,  Chap.  IX,  Trans.  D.  E.  Jones;   Bateman,  Elec- 
trical and  Optical  Wave  Motion,  p.  8. 


FROM   A   BENT   ANTENNA  5 

tion,  the  current  i  at  any  time  expressed  in  terms  of  I  the  maxi- 
mum current  must  be 

i  =  I  sin  pt  cos  -r—  (2) 

A 

where  X  =  the  wave  length  of  the  system, 


2x7. 
p  =  —  r—  is  the  angular  velocity 

A 


No 


w 


and  the  current  i  is  the  rate  of  change  of  the  charge  e  on  any 
doublet,  that  is 

de  di      d2e 

l  =  dt'  Tt=df 
Therefore 

d>e      di      2x71  2x7*        2x2 

^  =  ^  =  -r~cos^rcosir  (3) 

Substituting  (3)  in  (1)  we  get 

p/sinfl  /        r0—  zcos0\        2xz  , 

dE0a  =  t—yi  —  cos  p  (t  --       y       -  1  cos  -r-  dz        (4) 

in  which  -r—  is  given  the  shorter  form  p,  and  r0  —  z  cos  6  is 
A 

written  for  r  as  these  are  approximately  equal. 

The  doublet1  at  OPf  has  an  image  at  —  z  and  this  doublet 
will  have  an  effect  at  P  given  by  the  equation 

pi  sin  6  (         r  +  z  cos  6  \       2x2 

dEi  =  —  p  —  cos  p  [t  --       y       -   Icos  -y-  dz        (5) 

The  effect  due  to  the  two  doublets  on  the  point  P  is  found  by 
adding  (4)  and  (5),  and  the  action  of  all  the  doublets  and  their 

1  Theory  of  Images:  Jeans,  Electricity  and  Magnetism,  Chapter  VIII; 
Maxwell,  Electricity  and  Magnetism,  Chapter  XI;  Webster,  Electricity  and 
Magnetism,  p.  303. 


6  ELECTROMAGNETIC   OSCILLATIONS 

images,  by  integrating  from  0  to  h.     Then 

ind   Ch       2irz[        \     (         r0\   ,   pz  cos  6 
-  -y~ 

& cose 


The  terms  in  the  square  brackets  are  of  the  form  [Cos  (x  +  y) 
—  cos  (x  —  y)]  may  be  simplified  by  use  of  a  well-known  trigo- 
nometrical formula:  then 

2plsm8  /        r0\    Ch       pzcosd       2vz  , 

~^^—y — cos-r-ds    (7) 


The  integral  in  (7)  is  a  standard  form  and  is  found  in  any  table 
of  integrals.  Integrating  and  putting  in  the  limits  we  get  for 
the  integral  of  (7) 

fph  cos0      2Trh\  (  ,    2Trh\ 

in  I  — y r—  1       sm  I  ph  cos  0  H — r-  I 


This  reduces  to 


sin    -^-  (cos  0  —  1)         sin    -^  (cos 


~  (cos  0-1)  v  (cos  0+1) 

A  A 

Adding  these  fractions  we  get  for  the  integrated  term 

X  J 

«  —  ^rx  i  cos 

2?r  sm2  0[ 

Substituting  in  (7) 
27 


(2<ITh  _\     .      27T/1  /27T/J  \  27T/1 

I  "T~  cos  ^  )  sm  ^v  --  sln  I  "T~  cos  0  )  cos  "T~  cos  0 
\X  /         X  \X  /         X 


cos     ~~  cos 


\    . 

)  s 
/ 


Vhmt".  X  v  Iy  I        V  X 

-  cos  0  sin  (  ^^:  i  cos  z__-  L          (8) 


Equation  (8)  is  the  basis  for  the  following  brief  discussion 
of  a  vertical  antenna.     Poynting's  Vector  Theorem  states  that 


FROM   A   BENT   ANTENNA  7 

the  power  radiated  along  the  radius  at  any  point  of  a  sphere  is 
V/4:Tr(E  X  H)  per  unit  area.  In  the  case  of  the  vertical  E  =  H. 
The  energy  radiated  is  directly  proportional  to  E0.  The  wave 
length  is  equal  to  4&.  27/Fr0  is  constant.  The  average  value  of 
cos2  2ir/\(Vt  —  TQ)  is  |.  Therefore  the  expression  for  the  power 
radiated  at  an  angle  6  takes  the  simple  form 


(COS 

P.  =  K 


(fcosfl) 

'  (9) 


sin  6 

The  angle  at  which  the  greatest  amount  of  energy  is  radiated 
is  given  by 

fir  \ 

7p  cos  I  £  cos  6  \ 

0  n  T7 \      / 

dS  "  sin  B 

sin  (  ^  cos  0  )  ^  sin2  6  —  cos  (  ^  cos  8  }  cos  0 

— ^ /2 \_? ) . 

L  sm  6  ] 

The  solution  B  =  ir/2  makes  d2P0ldd2  negative,  and  therefore 
represents  the  direction  of  maximum  radiation  of  energy. 

Therefore  a  vertical  antenna  sends  out  its  maximum  energy 
along  the  horizontal.  Table  1  is  developed  from  (9),  K  being 
given  an  arbitrary  value.  Diagram  A  is  a  graph  of  Table  I, 
showing  the  approximate  energy  distribution.  The  distribution 
of  energy  is  such  that  nearly  all  the  radiation  lies  within  the  30° 
angle  with  the  horizon.  These  equations  (8)  and  (9)  throw  some 
light  on  the  action  of  tall  antennas.  The  radius  of  action  is 
increased  as  the  antenna  height  is  increased  for  two  reasons. 
First,  because  the  lengthening  of  the  antenna  automatically  in- 
creased the  wave  length  and  the  earth  is  a  better  conductor  for 
long  waves  than  for  short  ones.  Secondly,  because  the  waves 
from  a  high  antenna  are  not  brought  to  earth  so  quickly  (see  as- 
sumption three)  as  those  from  a  short  one.  Just  as  soon,  how- 
ever, as  the  wave  length  is  reached  for  which  the  earth  is  a 
perfect  conductor,  no  further  improvement  in  conduction  is 
effected  by  increasing  the  wave.  Similarly  there  is  a  limit  to 


8 


ELECTROMAGNETIC   OSCILLATIONS 


the  second  action,  so  that  there  is  no  reason  for  increasing  the 
height  of  the  sending  antenna  indefinitely. 


Angle 

0 
15 
30 
45 
60 
75 
90 


TABLE  I 

V 
0.00 

.078 

.144 

.422 

.672 

.902 
1.00 


*$ 

0.00 
.22 
.42 
.65 
.82 
.95 

1.00 


II.  THE  HORIZONTAL  PART 

In  order  to  apply  Hertz's  theory  to  the  horizontal  antenna, 
it  is  necessary  to  measure  the  co-latitude  from  the  axis  X.  The 
co-latitude  is  designated  by  \[/  and  the  azimuth  by  %•  The 
origin  remains  at  the  point  where  the  vertical  antenna  enters 
the  earth,  Fig.  3.  It  should  be  noticed  here  that  the  action  of 


FIG.  3 

the  horizontal  part  is  dependent  upon  the  length  of  the  vertical 
for  the  latter  determines  the  phase  of  the  current  at  any  point 
in  the  former.  Neither  in  theory  nor  in  practice  can  the  action 
of  the  horizontal  be  dissociated  from  the  vertical.  The  positions 
of  the  doublets  and  doublet  images  are  denoted  in  Fig.  4.  The 
small  letters  x,  y,  z  (or  x,  o,  h)  refer  to  the  doublets,  while  the 
position  of  the  point  in  space  P  is  given  by  large  X,  Y,  Z. 


FKOM    A   BENT   ANTENNA 


Hertz's  theory  may  now  be  applied  to  the  effect  at  P  of  the 
doublets  at  Q  and  Q'.    Then 


Now 


n  - 


=  V(X  -  z)2  +  p  +  (Z  + 


At  great  distances 


FIG.  4 


From  (11),  (12),  (13) 


-  2Zh 


2r 


-  2Xx 


As  before  in  (3) 


2r 


2irV  _        27T.  rr 
— - —  /  cos  -^-  \  Vt  — 


Also, 

w(*  r2^  (^  M 

fit  —  T7I=—  pi  COS  p  I   t  —  57   I 
\  V  )  \          V  J 


2ir 


cos 


27T 
cos    -— 


(10) 

(ID 
(12) 

(13) 


(14) 
(15) 


. 

(16) 


10  ELECTROMAGNETIC   OSCILLATIONS 


2ro  j  &»-j- (i+*)<fe    (17) 

Substituting  (16)  and  (17)  in  (10),  simplifying  by  the  trigono- 
metrical formula  for  the  sum  of  two  cosines;  neglecting  a;2  +  h2 
in  comparison  with  2Xx,  we  obtain  the  expression 

Jrt          ,T7             47r7  sin  \//      .    2irZh 
dEj,  =  dHy  =  — T^T —  •  sin  ~T — 

TQ  V  A  A7*o 

Vt  -  r0  +  x  cos  ^}  cosy  (h  +  x)dx  1     (18) 

The  integration  of  (18)  is  very  similar  to  that  of  (7)  and  gives 
finally 

21        .    2wZh  [  .    2ir  s  f        2nd 

E*  =     TZ   . — 7  sm-r sm-r-  (Vt  —  r0)  i  cos-^- 

r<>V  sm  ^         Xro    |_        A  X 

—  cos  (  -T—  cos  \l/  }  H-  cos  -r-  (Vt  —  r0) 

f                  2^  /2^  \11 

X  j  cos  ^  sm  -T- sm  I  -r—  cos  ^  1          (19) 

Upon  equations  (8)  and  (19)  are  based  all  the  conclusions  regard- 
ing a  bent  antenna.  The  total  flux  of  force  is  obtained  by  com- 
pounding EQ  of  (8)  with  the  E^  of  (19)  in  such  a  way  as  to  get 
the  complete  electric  intensity  E,  and  the  corresponding  magnetic 
intensity  H.  Although  the  doublets  which  give  rise  to  Ee  are 
perpendicular  to  those  producing  E$,  it  by  no  means  follows  that 
E#  and  Ee  are  at  right  angles;  in  fact,  they  make  an  angle  (a) 
with  one  another  which  varies  for  every  point  on  the  particular 
sphere  under  consideration.  Professor  Peirce,  of  Harvard  Uni- 
versity, has  shown  that  this  difference  in  direction  gives  rise  to 
a  third  term  in  the  power  radiation  { the  other  two  come  of  course 
from  (8)  and  (19) }  of  the  form 

sin  \{/  cos  8  cos  <j> 

2  cos  aE&E&,     where     cos  a.  =  —  1 r-^-r x— 

1  —  sm2  8  cos2  4> 

The  average  value  of  this  term,  when  multiplied  by  F/47T,  he 


FROM   A   BENT  ANTENNA 

calls  the  mutual  power.     In  the  problem  under  discussion 
—  2K  cos  0  cos  4>        .    2irhZ  J 


11 


4?r 


X 


sm 


sin  0(1  —  sin2  0  cos  <f>) ""  L   \rQ    [ 
—  sin  ( 


cos     sin 


cos 


2irh        2wh  cos  0 
^:—  cos—  —  T  ---  cos 
A  A 


2irh   .    ir2h  cos  (9  .     /rtm 
-r-  sm  -  r-  —  f    (20) 
A  A 


where  for  convenience  K  is  put  equal  to  72/7rFr02.  This  equation 
is  true  only  for  the  average  value  of  (V/2ir)  cos  aEeE^  so  that 
the  term 

cos2  -T-  (Vt  —  r0)  =  \  and  sin  —  (Vt  —  r0)  cos  r-  (Vt  —  r0)  =  0 

A  A  A 

SECOND  PART— NUMERICAL  CALCULATION 

The  power  radiated  per  unit  area  in  certain  zones  at  distances 
2,000,  6,000  and  9,000  meters  above  the  earth's  surface,  and  at 
a  distance  10,000  meters  from  the  origin  will  now  be  calculated 
by  means  of  equations  (8),  (19)  and  (20).  In  these  equations 
let  h  =  10  meters,  d  =  100  meters,  X  =  4(&  +  d)  =  440.  Then 
the  power  radiated  from  the  vertical  part  may  be  expressed  by 
(8)  in  the  form : 

V  E? 

47T 

=  K  jcosf  -r-  COS0J  sin-r- cos  0  sin  (  ~-  cos  0)  cos  ~^- 


10,000  M. 


jwoo  jr. 

"^6000  M., 

ASOOOM. 


FIG.  5 


The  values  in  Table  II  are  calculated  from  this  equation: 


12 


ELECTROMAGNETIC  OSCILLATIONS 


X  =  440M 
h  =  10M 
r  =  10000M 


VERTICAL — TABLE  II 

Height 

9000M 
6000M 

5000 
2000 
0 


.0049K 
.0160 

.0164 
.0196 
.0196 


The  calculations  of  E92(V/4:ir)  for  the  horizontal  part  are  by  no 
means  as  simple  as  this.  The  value  of  the  angle  in  formula  (19) 
changes  for  every  point  on  the  zone  at  the  height  9000,  6000 
and  so  on.  This  change  must  be  calculated.  In  Fig.  6  ABC 


FIG.  6 

is  a  spherical  triangle  with  sides  8,  \l/,  irj '2  and  angle  <£  given. 
Then  from  a  well-known  formula  in  the  trigonometry  of  triangles: 

Cos  \l/  =  sin  6  cos  <j> 

The  power  radiated  from  the  horizontal  antenna  through  the 
zone  Z  =  9000,  6000,  etc.,  will  vary  for  different  angles  of  <j>, 
that  is,  the  bent  part  of  the  antenna  has  a  directive  effect  on  the 
oscillations. 

In  equation  (19) 

5in,  f  2rZh  } 

VW       „         I    Xrn    I  f         2nd  2ird 

j  cos2  -T- — r  1  +  cos2  \l/  sin2  -r— 


47T 


=  K 


sin2  \l/ 


2ird 
—  2  cos  -r—  cos 


\ 

1  —  2 


cos  1  —  2  cos  \f/  sin 


sm 


cos 


XX 

From  this  equation  the  values  of  Table  III  are  calculated  for 
the  different  zones  and  different  directions.  The  operations  are 
long  and  tedious  but  not  difficult.  The  horizontal  part  of  the 


FROM  A  BENT  ANTENNA 


13 


antenna  has  a  slight  directive  effect  perpendicular  to  the  direc- 
tion in  which  it  points.  This  is  contrary  to  what  one  would 
expect:  because  all  experiments  show  a  directive  effect  in  the 
direction  away  from  the  free  end  of  the  antenna.  However,  it 
has  been  shown  by  Pocklington  that  a  circular  wire  radiates 
more  power  perpendicular  to  its  area  than  in  any  other  direction 
and  the  form  of  equation  (19)  shows  that  Pocklington's  method 
of  doublets  applies  to  this  problem  and  that  the  solution  is 
correct  to  the  approximations  made. 

HORIZONTAL  PART — TABLE  III 


Height 

<£ 

7?  2  V   • 

^5 

0 

9000 

0 

.0097 

180 

30 

.0094 

150 

45 

.0092 

135 

60 

.0097 

120 

90 

.0107 

90 

6000 

0 

.0022 

180 

30 

.0028 

150 

45 

.0031 

135 

60 

.0040 

120 

90 

.0047 

90 

2000 

0 

.0004 

180 

30 

.0001 

150 

60 

.0003 

120 

90 

.0000 

90 

It  will  now  be  shown  that  the  power  arising  from  the  mutual 
effect  (eq.  20)  tends  to  modify  the  directive  effect  at  high  points 
in  the  atmosphere  in  such  a  way  as  to  give  a  fore  and  aft  directive 
effect. 

The  values  of  Table  IV  are  calculated  from  equation  (20). 

MUTUAL  EFFECT — TABLE  IV 


Height. 

0 

5^ 

47, 

4> 

9000 

0 

.0027K 

180 

30 

.0022 

150 

45 

.0014 

135 

60 

.0005 

120 

90 



90 

6000 

0 

.0030 

180 

30 

.0023 

150 

45 

.0015 

135 

60 

.0009 

120 

90 



90 

2000 

Negligible 

14 


ELECTEOMAGNETIC   OSCILLATIONS 


Adding  up  the  vertical,  the  horizontal  and  the  mutual  effects 
contained  in  Tables  II,  III  and  IV,  we  obtain  the  complete 
power  radiated  through  unit  area  of  the  zones.  The  results  are 
given  in  Table  V,  in  which  K  has  been  set  equal  to  an  arbitrary 
value.  Plate  B  is  plotted  from  Table  V.  In  Plate  C,  the  dis- 
tribution at  9000  M  is  compared  to  a  curve  obtained  experi- 
mentally by  Fleming. 

TOTAL  EFFECT — TABLE  V 


Height 

0 

p 

4> 

9000 

0 

.0173 

180 

30 

.0165 

150 

45 

.0155 

135 

60 

.0151 

320 

90 

.0156 

90 

6000 

0 

.0212 

180 

30 

.0211 

150 

45 

.0206 

135 

60 

.0209 

120   . 

90 

.0207 

90 

2000 

0 

.0206 

180 

30 

.0197 

150 

60 

.0199 

120 

90 

.0196 

90 

The  symmetry  of  the  curves  developed  from  the  theory  can 
be  reconciled  with  the  asymmetrical  curve  found  in  the  experi- 
ments by  supposing  that  the  electrical  waves  brought  back  to 
the  earth  from  high  in  the  atmosphere  are  more  intense  toward 
the  bend  in  the  antenna  than  at  the  free  end.  Zenneck  (Zenneck, 
Phys.  Zeitsch.,  Vol.  9,  p.  553,  1908)  has  shown  that  this  difference 
may  be  due  to  imperfect  conductivity  in  the  earth's  surface. 

SUMMARY 

First :  The  equations  developed  by  Pocklington,  Abraham 
and  Peirce  have  been  applied  to  an  antenna  with  vertical  and 
horizontal  parts  in  such  a  way  as  to  find  the  energy  given  out 
for  the  fundamental  vibration: 

X  =  4(fc  +  d) 

Second:  The  intensities  so  obtained  are  plotted  and  are  shown 
to  have  a  fore  and  aft  directive  inclination.  The  intensity  fore 
and  aft  is  symmetrical  and  not  asymmetrical  as  required  by  the 
experiments. 


FROM   A   BENT   ANTENNA  15 

Third:  Close  to  the  antenna  the  intensities  are  symmetrical 
in  azimuth  agreeing  with  experiment. 

Fourth:  The  conclusion  is  that  the  fore  and  aft  asymmetry 
of  a  bent  antenna  is  caused  by  the  difference  in  conductivity 
between  the  atmosphere  and  the  earth  for  the  electromagnetic 
oscillation. 

Fifth:  The  forms  of  the  resulting  equations  show  that  Peirce's 
assumptions  regarding  the  current  satisfy  Pocklington's  criterion 
for  the  use  of  Maxwell's  Equations. 

ADDENDUM 

The  Integration  of  Equations  (18),  p.  10. 
The  integrable  part  of  (18)  is 

I  sin  —  {  Vt  —  rQ  -f-  x  Cos  \//}  cos  -r-  (h  +  x)dx 

Expand  and  multiply,  thus: 

n.    27T.T.  27rxCost  ,         27r.__  .   27rxCost~] 

sin-r-  (Vt—  r0)  cos  --  r  --  +  cos—  (Vt—  r0)  sin  --  — 
A  A  A  A 

2irh  sin  2 


[2irh        2irx  2ir 

cos-^cos^    -sm  — 

r2ir  ,.  2wh        2irx  Cos  \[/        2irx  . 

sm—  (Vt  —  rQ)  cos^r—  cos  -  r  -  COS-T—  ax    (21) 
A  A  A  A 

2ir  „  2wh   Cd  .    2irx  Cos  \l/        2irx  , 

+  cos  -r-  (Vt  —  fo)  cos  -r-        sm  -  r  -  cos  -T"  dx  (22) 

A  A    ,/Q  A  A 


.    2ir  .    27rh  Cd       2wx  Cos  ^  .    2wx 

—  sm^r-  (Vt  —  TQ)  sin^r—        cos  --  7—  -  sm-^r-  dx  (23) 

A  A    J0  A  A 


27r/T_               .    2irh  r  .    2jrxCos\l/  .    2wx 
—  cos  -T-  (p  t  —  r0)  sm  -r-        sm r—   -  sm  —  dx  (24) 

A  A    t/o  A.  A 

The  integration  of  (21)  is  the  same  as  that  of  (7)  on  page  4  and 
gives 

X  2wdcos\l/  .    2wd  .    2wdcos\l/^     27rd~\ 

2ir  sin2  \//  \_  X  X  X  X  J 

The  integration  of  the  integral  part  (22)  comes  under  Form  360 


16 


ELECTROMAGNETIC   OSCILLATIONS 


in  Peirce's  Table  of  Integral  and  simplifies  into 


2wd 


cos 


. 
+  sin 


.    2irh 
sin  —  -  cos 


(26) 


(23)  comes  under  form  (360)  but  in  a  different  way  from  (22). 
The  complete  integration  follows: 
To  find 

2irx       2irx  Cos 


r 


sin  —  cos 


(Form  360) 


sin  mx  cos  nx  ax  =  — 


cos  (m  —  n)x      cos  (m  +  ri)x 


Let 


2(m  -  n) 
2ir  2ir  cos 


2(m 


Then  integration  gives 


47T 


2w  sin' 


_X 

47T 


27T(Z  COS  \I/ 
(1  +  COS  \l/)  \  COS  —  COS  --  r  - 
A   ,  A 

2nd  .    2irdcos\l/ 
+  sm  ^—  sin  -  —  T— 

A  A 

2nd       2ird  cos  ^ 
(1  —  cos  \p)  \  cos  -r-  cos  -  -  - 
A  A 

2wd  .    2ird  cos  \1/ 


—  sm  -r—  sin 
A 


2ir  sin2 


FROM   A   BENT  ANTENNA  17 

2ird        2ird  cos  , 

(27) 


2ir  sin2  \p  .    2ird  .    2nd  cos  ^ 

+  cos  \f/  sin  -r—  sin r— 1 

A  A 

The  integrable  part  of  (24) 

r2wx  Cos  \l/  .    2irx  , 
sin r sin  —  dx 
A  A 

is  also  a  standard  form  and  becomes  finally 


2ird  cos  \l/  .    2ird        .    2ird  cos  \L       2nd  ] 
>s^cos r sm^r sm COS-T—  )  (28) 

A  A  A  A     J 

Substitute  25,  26,  27  and  (28)  in  (21),  (22),  (23)  and  (24).    Then 
•     insert  the  result  in  (18)  and  obtain: 

—  4wl  sin  \//  .    27rZh          X 

•f-^Ji    ~~~  Tr-\  Sin      »  •   7%  .     a     T 


,    27T                         2irh  I        2nd  cos  ^  .    2ird 
sm  -T-  (Vt  —  rQ)  cos  ^—  j  cos sin  ^~ 


,    2nd  Cost       Ivd] 
—  cos  \l/  sin cos  -r—  f 

A  A     J 


27r/T.  2irh  I        2nd  cost 

COS  y  (Vt  —  r0)  COS  -r—  j  COS ^ COS 


.    2,ird  cos  \[/  .    2wh 
+  sin  --  r-  --  sin  —  --  cos  \[/  f 


,    2<jr  .__  ,    2irh  f        2<7rd       2ird  cos 

+  sin  —  (Vt  —  r0)  sin  -r—  j  cos  -r-  cos  --  r— 


2ird  .    2ird  cos 
+  cos  t  sin  ~"  sin  --  r— 


„  1 

2ir  ,-  .    2irh  f  ,  cos  2nd  cos  \l/  .    2wd 

+  cos  -:-  (Vt  —  TO)  sin  ^—  j  —  cos  \l/  —     ~Y~     ~  sm  "T~ 

{~] 
| 


,    2ird  cos  t       2<jrd 
sm  --  ^  --  COS~X 


18  ELECTROMAGNETIC   OSCILLATIONS 

Now 

.       X      2irh      TT      2wd 
=  4'''T~    =  2'T~ 
so  that 

2irh  2nd 

cos  -T—  =  sin  -T— 

A  A 

2ird 


sin  -r—  =  cos  -r— 

A  A 

When  these  expressions  are  substituted  in  (29),  the  terms  in 
sin  (2irl\)(Vt  —  TO)  and  those  in  cos  (2ir/\)(Vt  —  r0)  may  be 
added  together  and  give  the  simple  form: 

21         ,    27rZh[  .    27r  f        2wd  cos  ^ 


.-  sin  -T  —      sin—  L      —  TO)     cos 


T7    .     . 
r0V  sin  ^         Xr0    L 


1  27r.T.  . 

—  cos  -r—  f  +  cos  -r-  (  Vt  —  r0)  j  —  cos  \f/  sin 

A     J  A 


which  is  equation  (19),  page  (9). 


/ 


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